EGO with a failure region

from __future__ import annotations

import numpy as np
import tensorflow as tf

np.random.seed(1234)
tf.random.set_seed(1234)

The problem

This notebook is similar to the EI notebook, where we look to find the minimum value of the two-dimensional Branin function over the hypercube \([0, 1]^2\). But here, we constrain the problem, by adding an area to the search space in which the objective fails to evaluate.

We represent this setup with a function masked_branin that produces null values when evaluated in the disk with center \((0.5, 0.4)\) and radius \(0.3\). It’s important to remember that while we know where this failure region is, this function is a black box from the optimizer’s point of view: the optimizer must learn it.

import trieste


def masked_branin(x):
    mask_nan = np.sqrt((x[:, 0] - 0.5) ** 2 + (x[:, 1] - 0.4) ** 2) < 0.3
    y = np.array(trieste.objectives.branin(x))
    y[mask_nan] = np.nan
    return tf.convert_to_tensor(y.reshape(-1, 1), x.dtype)

As mentioned, we’ll search over the hypercube \([0, 1]^2\) …

from trieste.space import Box

search_space = Box([0, 0], [1, 1])

… where the masked_branin now looks as follows. The white area in the centre shows the failure region.

from util.plotting_plotly import plot_function_plotly

fig = plot_function_plotly(
    masked_branin, search_space.lower, search_space.upper, grid_density=70
)
fig.update_layout(height=400, width=400)
fig.show()

Define the data sets

We’ll work with two data sets

• one containing only those query_points and observations where the observations are finite. We’ll label this with OBJECTIVE.

• the other containing all the query points, but whose observations indicate if evaluating the observer failed at that point, using 0 if the evaluation failed, else 1. We’ll label this with FAILURE.

Let’s define an observer that outputs the data in these formats.

OBJECTIVE = "OBJECTIVE"
FAILURE = "FAILURE"


def observer(x):
    y = masked_branin(x)
    mask = np.isfinite(y).reshape(-1)
    return {
        OBJECTIVE: trieste.data.Dataset(x[mask], y[mask]),
        FAILURE: trieste.data.Dataset(x, tf.cast(np.isfinite(y), tf.float64)),
    }

We can evaluate the observer at points sampled from the search space.

num_init_points = 15
initial_data = observer(search_space.sample(num_init_points))

Build GPflow models

We’ll model the data on the objective with a regression model, and the data on which points failed with a classification model. The regression model will be a GaussianProcessRegression wrapping a GPflow GPR model, and the classification model a VariationalGaussianProcess wrapping a GPflow VGP model with Bernoulli likelihood. The GPR and VGP models are build using Trieste’s convenient model build functions build_gpr and build_vgp_classifier. Note that we set the likelihood variance to a small number because we are dealing with a noise-free problem.

from trieste.models.gpflow import build_gpr, build_vgp_classifier

regression_model = build_gpr(
    initial_data[OBJECTIVE], search_space, likelihood_variance=1e-7
)
classification_model = build_vgp_classifier(
    initial_data[FAILURE], search_space, noise_free=True
)

Build Trieste models

We now specify how Trieste will use our GPflow models within the BO loop.

For our VGP model we will use a non-default optimization: alternate Adam steps (to optimize kernel parameter) and NatGrad steps (to optimize variational parameters). For this we need to use the BatchOptimizer wrapper and set the use_natgrads model argument to True in our VariationalGaussianProcess model wrapper.

We’ll train the GPR model with the default Scipy-based L-BFGS optimizer, and the VGP model with the custom algorithm above.

from trieste.models import TrainableProbabilisticModel
from trieste.models.gpflow.models import (
    GaussianProcessRegression,
    VariationalGaussianProcess,
)
from trieste.models.optimizer import BatchOptimizer


models: dict[str, TrainableProbabilisticModel] = {
    OBJECTIVE: GaussianProcessRegression(regression_model),
    FAILURE: VariationalGaussianProcess(
        classification_model,
        BatchOptimizer(tf.optimizers.Adam(1e-3)),
        use_natgrads=True,
    ),
}

Create a custom acquisition function

We’ll need a custom acquisition function for this problem. This function is the product of the expected improvement for the objective data and the predictive mean for the failure data. We can specify which data and model to use in each acquisition function builder with the OBJECTIVE and FAILURE labels. We’ll optimize the function using EfficientGlobalOptimization.

from trieste.acquisition.rule import EfficientGlobalOptimization
from trieste.acquisition import (
    SingleModelAcquisitionBuilder,
    ExpectedImprovement,
    Product,
)


class ProbabilityOfValidity(SingleModelAcquisitionBuilder):
    def prepare_acquisition_function(self, model, dataset=None):
        def acquisition(at):
            mean, _ = model.predict_y(tf.squeeze(at, -2))
            return mean

        return acquisition


ei = ExpectedImprovement()
pov = ProbabilityOfValidity()
acq_fn = Product(ei.using(OBJECTIVE), pov.using(FAILURE))
rule = EfficientGlobalOptimization(acq_fn)  # type: ignore

Run the optimizer

Now, we run the Bayesian optimization loop for twenty steps, and print the location of the query point corresponding to the minimum observation.

bo = trieste.bayesian_optimizer.BayesianOptimizer(observer, search_space)

num_steps = 20
result = bo.optimize(
    num_steps, initial_data, models, rule
).final_result.unwrap()

arg_min_idx = tf.squeeze(
    tf.argmin(result.datasets[OBJECTIVE].observations, axis=0)
)
print(f"query point: {result.datasets[OBJECTIVE].query_points[arg_min_idx, :]}")

WARNING:tensorflow:5 out of the last 20 calls to <function expected_improvement.__call__ at 0x7faa84a27050> triggered tf.function retracing. Tracing is expensive and the excessive number of tracings could be due to (1) creating @tf.function repeatedly in a loop, (2) passing tensors with different shapes, (3) passing Python objects instead of tensors. For (1), please define your @tf.function outside of the loop. For (2), @tf.function has experimental_relax_shapes=True option that relaxes argument shapes that can avoid unnecessary retracing. For (3), please refer to https://www.tensorflow.org/guide/function#controlling_retracing and https://www.tensorflow.org/api_docs/python/tf/function for  more details.
Optimization completed without errors
query point: [0.96267202 0.17098896]

We can visualise where the optimizer queried on a contour plot of the Branin with the failure region. The minimum observation can be seen along the bottom axis towards the right, outside of the failure region.

import matplotlib.pyplot as plt
from util.plotting import plot_gp_2d, plot_function_2d, plot_bo_points

mask_fail = (
    result.datasets[FAILURE].observations.numpy().flatten().astype(int) == 0
)
fig, ax = plot_function_2d(
    masked_branin,
    search_space.lower,
    search_space.upper,
    grid_density=50,
    contour=True,
)
plot_bo_points(
    result.datasets[FAILURE].query_points.numpy(),
    ax=ax[0, 0],
    num_init=num_init_points,
    mask_fail=mask_fail,
)
plt.show()

We can also plot the mean and variance of the predictive distribution over the search space, first for the objective data and model …

from util.plotting_plotly import (
    plot_model_predictions_plotly,
    add_bo_points_plotly,
)

arg_min_idx = tf.squeeze(
    tf.argmin(result.datasets[OBJECTIVE].observations, axis=0)
)

fig = plot_model_predictions_plotly(
    result.models[OBJECTIVE],
    search_space.lower,
    search_space.upper,
)
fig = add_bo_points_plotly(
    x=result.datasets[OBJECTIVE].query_points[:, 0].numpy(),
    y=result.datasets[OBJECTIVE].query_points[:, 1].numpy(),
    z=result.datasets[OBJECTIVE].observations.numpy().flatten(),
    num_init=num_init_points,
    idx_best=arg_min_idx,
    fig=fig,
    figrow=1,
    figcol=1,
)

fig.show()

… and then for the failure data and model

fig, ax = plot_gp_2d(
    result.models[FAILURE].model,  # type: ignore
    search_space.lower,
    search_space.upper,
    grid_density=50,
    contour=True,
    figsize=(12, 5),
    predict_y=True,
)

plot_bo_points(
    result.datasets[FAILURE].query_points.numpy(),
    num_init=num_init_points,
    ax=ax[0, 0],
    mask_fail=mask_fail,
)

plot_bo_points(
    result.datasets[FAILURE].query_points.numpy(),
    num_init=num_init_points,
    ax=ax[0, 1],
    mask_fail=mask_fail,
)

plt.show()

LICENSE

Apache License 2.0